Properties

Label 6.23_10337.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 23 \cdot 10337 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$237751= 23 \cdot 10337 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.23_10337.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 30 a + 17 + \left(16 a + 20\right)\cdot 41 + \left(17 a + 22\right)\cdot 41^{2} + \left(14 a + 22\right)\cdot 41^{3} + \left(13 a + 8\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 34 a + 37 + \left(18 a + 38\right)\cdot 41 + \left(8 a + 25\right)\cdot 41^{2} + \left(17 a + 13\right)\cdot 41^{3} + \left(3 a + 29\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 35 + 31\cdot 41 + 28\cdot 41^{2} + 33\cdot 41^{3} + 6\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 a + 16 + \left(22 a + 20\right)\cdot 41 + \left(32 a + 32\right)\cdot 41^{2} + \left(23 a + 15\right)\cdot 41^{3} + \left(37 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 22 + 36\cdot 41 + 29\cdot 41^{2} + 14\cdot 41^{3} + 24\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 13 + 16\cdot 41 + 7\cdot 41^{2} + 15\cdot 41^{3} + 38\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 11 a + 25 + \left(24 a + 40\right)\cdot 41 + \left(23 a + 16\right)\cdot 41^{2} + \left(26 a + 7\right)\cdot 41^{3} + \left(27 a + 34\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation
$(1,2,3,4,5,6,7)$
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$6$
$21$$2$$(1,2)$$4$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$210$$4$$(1,2,3,4)$$2$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$1$
$210$$6$$(1,2,3)(4,5)(6,7)$$-1$
$420$$6$$(1,2,3)(4,5)$$1$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$-1$
$504$$10$$(1,2,3,4,5)(6,7)$$-1$
$420$$12$$(1,2,3,4)(5,6,7)$$-1$
The blue line marks the conjugacy class containing complex conjugation.